### A boundary Harnack principle for infinity-Laplacian and some related results.

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Let $A$ be a closed set of $M\simeq {\mathbb{R}}^{n}$, whose conormai cones $x+{y}_{x}^{*}\left(A\right)$, $x\in A$, have locally empty intersection. We first show in §1 that $\text{dist}\left(x,A\right)$, $x\in M\setminus A$ is a ${C}^{1}$ function. We then represent the n microfunctions of ${\mathcal{C}}_{A|X}$, $X\simeq {\mathbb{C}}^{n}$, using cohomology groups of ${\mathcal{O}}_{X}$ of degree 1. By the results of § 1-3, we are able to prove in §4 that the sections of ${\left.{\mathcal{C}}_{A|X}\right|}_{{\dot{\pi}}^{-1}\left({x}_{0}\right)}$, ${x}_{0}\in \partial A$, satisfy the principle of the analytic continuation in the complex integral manifolds of ${\left\{H\left({\varphi}_{i}^{C}\right)\right\}}_{i=1,\mathrm{\dots},m}$, $\left\{{\varphi}_{i}\right\}$ being a base for the linear hull of ${\gamma}_{{x}_{0}}^{*}\left(A\right)$ in ${T}_{{x}_{0}}^{*}M$; in particular we get ${\left.{\mathrm{\Gamma}}_{A\times {}_{M}T{}^{*}{}_{M}X}\left({\mathcal{C}}_{A|X}\right)\right|}_{\partial A\times {}_{M}\dot{T}{}^{*}{}_{M}X}=0$. When $A$is a half space with ${C}^{\omega}$-boundary,...

In the present paper we study the unique solvability of two non-local boundary value problems with continuous and special gluing conditions for parabolic-hyperbolic type equations. The uniqueness of the solutions of the considered problems are proven by the “abc” method. Existence theorems for the solutions of these problems are proven by the method of integral equations. The obtained results can be used for studying local and non-local boundary-value problems for mixed-hyperbolic type equations...

MSC 2010: 44A35, 44A40

We prove that homogeneous problem for PDE of second order in time variable, and generally infinite order in spatial variables with local two-point conditions with respect to time variable, has only trivial solution in the case when the characteristic determinant of the problem is nonzero. In another, opposite case, we prove the existence of nontrivial solutions of the problem, and we propose a differential-symbol method of constructing them.